Field-controlled conical intersections in the vortex lattice of quasi 2D pure strongly type-II superconductors at high magnetic fields
FField-controlled conical intersections in the vortex lattice of quasi 2D pure stronglytype-II superconductors at high magnetic fields
T. Maniv ∗ and V. Zhuravlev Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel (Dated: November 19, 2018)It is shown that the Dirac fermion structures created in the middle of the Landau bands in thevortex-lattice state of a pure 2D strongly type-II superconductor at half-integer filling factors canbe effectively controlled by the external magnetic field. The resulting field-induced modulation ofthe magneto-oscillations is shown to arise from Fermi-surface resonance scattering in the vortexcore regions. Possible observation of the predicted effect in a quasi 2D organic superconductor isdiscussed.
PACS numbers: 74.25.Ha, 74.25.Uv, 74.78.-w,74.70.Kn
In a pure strongly type-II superconductor under a uni-form magnetic field the quasi particle spectrum is gap-less in a broad field range below the upper critical field H c [1],[2],[3]. In this field range scattering of quasi par-ticles by the vortex lattice interferes with the Landauquantization of the electron motion perpendicular to themagnetic field to form magnetic (Landau) Bloch’s bands.The physical picture is of an extended Bloch state whichbreaks magnetically into localized cyclotron orbits [3]. Inpure 2D, or quasi 2D superconductors, such as e.g. theorganic charge transfer salt κ − ( ET ) Cu ( SCN ) [4],under a magnetic field perpendicular to the easy con-ducting planes, the underlying normal electron spectrumis fully quantized and the effect of the vortex lattice isvery pronounced. Furthermore, due to the suppressed en-ergy dispersion along the magnetic field direction and theparticle-hole symmetry inherent to the superconducting(SC) state, the quasi particle spectrum exhibits peculiarfeatures that are missing in the 3D case. For example, atdiscrete magnetic field values where the chemical poten-tial is located in the middle of a Landau band, so that theunderlying normal state spectrum satisfies particle-holesymmetry, the calculated quasi-particle density of states(DOS) has a Dirac Fermion structure [5], which reflectstopological singularities at the vortex lattice cores.In the present paper we reveal a physical mechanismwhich controls the Dirac fermion structures created in themagnetic Brillouin zone (BZ) of the vortex lattice of 2Dstrongly type-II superconductors at high magnetic fields,and discuss possible experimental probes of their appear-ance. The ability to create and control Dirac fermionsjust by varying an external parameter (magnetic field inour case) is of great importance for future technologicalapplications (see, e.g. [6],[7]). For the present analysis weconsider a model of a 2D electron system under a perpen-dicular uniform magnetic field H = (0 , , H ), neglecting,for the sake of simplicity, the Zeeman spin splitting andassuming a singlet, s -wave electron pairing. It shouldbe noted that in a 2D or quasi 2D electron system thecondition of zero spin-splitting can be realized by tilt-ing the magnetic field direction with respect to the easy conducting planes [4, 8].The corresponding equations for the quasi particlestates in the mean-field approximation are the wellknown Bogoliubov de Gennes (BdG) equations in theLandau orbitals representation [1, 9]: X n ∆ n,n ( k ) v n ( k ) = ( ε − ξ n ) u n ( k ) , X n ∆ ∗ n ,n ( k ) u n ( k ) = ( ε + ξ n ) v n ( k ) , (1)where the single-electron energy measured relative to thechemical potential µ is given by ξ n = ~ ω c ( n − n F ) , n =0 , , , ... , n F = µ/ ~ ω c − /
2, and ω c = eH/m ∗ c is theelectronic cyclotron frequency. The matrix, ∆ n,n ( k ), isdiagonal in the magnetic Brillouin zone, but non-diagonalin the Landau-level (LL) indices n, n . The pair potential∆ ( r ) should,in principle,be determined self consistentlywith the eigenfunctions v n ( k ) , u n ( k ) [3]. It will be veryhelpful to avoid the complexity involved in a fully self-consistent approach by assuming ∆ ( r ) to have the formof a vortex lattice, as will be elaborated below (see alsoSM2 [10]). Since the Abrikosov vortex lattice shares withthe self consistent pair-potential in the lowest LL approx-imation [3] the feature of main interest here (i.e. thetopological singularity of the vortex-lattice cores) this isa reasonable assumption.A very important information is encoded in thesematrix elements[1]: The zeros of the diagonal groundLL matrix-element ∆ , ( k ), are all of the first or-der and form a lattice dual to the Abrikosov vortexlattice rotated by 90 ◦ . Matrix elements (diagonalas well as off-diagonal) with higher LL indices havealso zeros of higher orders. In the diagonal LL ap-proximation, which is valid sufficiently close to H c ,and in the case where n F coincides with a LL in-dex, n = n F (i.e. corresponding to particle-holesymmetry in the normal state), one trivially solvesthe BdG equations to find the quasi-particle energies: ε ± ( k ) = ± | ∆ n,n ( k ) | , and the corresponding eigen-states: ( u n ± ( k ) , v n ± ( k )) = (cid:0) ± e iφ ( k ) / , e − iφ ( k ) / (cid:1) / √ a r X i v : . [ c ond - m a t . s up r- c on ] M a y D n F (cid:72) Ε(cid:142) (cid:76) n F (cid:61) Ε(cid:142) n F (cid:61) . n F (cid:61) (cid:45) (cid:45) FIG. 1: Quasi-particle DOSs, as functions of e ε ≡ ε/ ~ ω c , cal-culated by solving the BdG equations (1) near the Fermi en-ergy ( ε = 0) for integers n F (= 57 ,
58) and a half integer n F (= 58 .
5) with ∆ / ~ ω c = 1 . Note the sharp increase of theDirac-shaped DOS slop from n F = 57 to 58. with: e iφ ( k ) = ∆ n,n ( k ) / | ∆ n,n ( k ) | . Thus, near eachvortex core, k j , in the reciprocal vortex-lattice [1], where | ∆ n,n ( k ) | → η n | k − k j | →
0, the corresponding quasi-particle dispersion relation exhibits a conical intersec-tion of the ± branches at the chemical potential (zeroenergy), in close similarity to graphene Dirac cone struc-ture on a single honeycomb sub-lattice [11],[12]. Tak-ing into account off-diagonal LL pairing, which is cru-cial at magnetic fields even slightly away from H c , re-quires numerical solution of Eq.(1) with a great loss ofphysical insights. However, the topological nature of thevortex cores singularity indicates that significant finger-prints of this singularity should appear in the disper-sion relation under the influence of off-diagonal LL pair-ing. The results for the quasi-particle density of states(DOS), shown in Fig.(1), support this conjecture: Eachbroadened LL splits into two sub-bands due to Andreevscattering of quasi particle-quasi hole in the vortex re-gions (see e.g. [13]). The split sub-bands join to formstraight-line intersection whenever the chemical poten-tial is located in the middle of a Landau band and thefilling factor of the Landau levels, µ/ ~ ω c = n F + 1 / n F is integer). The linear vanishing of thequasi-particle DOS at the Fermi energy (see Fig.1) is aconsequence of avoiding crossings, close to conical inter-sections, of quasi-particle energy branches at many wavevectors in the 2D magnetic Brillouin zone (see SM 1). Atnon-integer n F , where the two (particle-hole) branches ofthe quasi-particle energy in the diagonal approximationdo not intersect, the exact energy spectrum develops agap between the two branches, ± | ∆ n,n ( k ) | , with a verysmall number of states arising from off-diagonal LL pair-ing occupying the ’diagonal gap’. Fig.(1) illustrates thesituation for a half integer n F , where particle-hole sym-metry is satisfied in both the normal and SC states. For any other non-integer n F value this symmetry is obeyedin the SC state, but not in the normal state. At inte-ger values of n F normal-electron Landau tubes cross the(cylindrical) Fermi surface and the oscillatory magnetiza-tion has maxima, whereas the minima of the oscillationsoccur at the center of the cyclotron gaps (i.e. at half inte-gers n F ). The gaps developed in the quasi-particle spec-trum in the vortex state at half integers n F ensure thatthe minima of the oscillations also occur at half-integervalues of n F . The envelope of the magneto-quantum os-cillations in the vortex state is therefore controlled byDirac-shaped quasi-particle DOS at integer values of n F .To reveal the physical mechanism controlling the fielddependence of the DOS one must complement the nu-merical approach of the BdG equations by resortingto an analytical method that is sensitive to quasi par-ticle (Andreev) scattering in the vortex core regions.The exact Gorkov-Ginzburg-Landau perturbation ap-proach developed in Ref.([14]) possesses the requiredsensitivity. Following Ref.([14]) the leading order termin the SC thermodynamic potential, that is sensitiveto vortex structure, i.e. the quartic term, takes theform: Ω = N ( k B T / e ∆ P ∞ ν =0 I ν , where N is thetotal number of flux lines threading the SC sample, e ∆ = (∆ / ~ ω c ) , and I ν is written as a 4D ’tempo-ral’ integral I ν = Q j =1 R ∞ dτ j ( β ( γ ) /α ) e − $ ν τ + − in F τ − .In this expression τ + = P j =1 τ j , τ − = P j =1 ε j τ j , ε j ≡ ( − j +1 , α = P j =1 α j , α j ≡ − e iε j τ j , β ( γ ) = P G h e − θ − | G | / (1 + γ ) + e − θ + | G | / (1 − γ ) i / γ = ( α α − α α ) /α , θ − = (1 − γ ) / (1 + γ ),and θ + = (1 + γ ) / (1 − γ ). In β ( γ ) the summation isover the reciprocal vortex-lattice vectors, G , measuredin units of the inverse magnetic length a − H = p eH/c ~ , $ ν ≡ ω ν /ω c , and ω ν = (2 ν + 1) πk B T / ~ , ν = 0 , ± , ... isthe Matsubara frequency at temperature T . The ampli-tude of the SC order parameter, ∆ = S − R d r | ∆( r ) | ,with S = N πa H , is treated as a variational parameter forminimizing the thermodynamic potential Ω SC (∆ ) . Thesalient features of I ν are: (1) The simple oscillatory fac-tor e − in F τ − , revealing directly the Fourier transformedcomponents with respect to the dHvA frequency F = n F H . (2) The vortex lattice structure factor β ( γ )- anextension of the well known Abrikosov parameter to thehigh-field regime, which depends on the electronic ‘tem-poral’ variables, τ j , through a single composite variable- γ , and on the vortex structure through the reciprocalvortex lattice vectors G . Remarkably, approaching thepoints γ = +1 , − ± γ ) vanishes in the denom-inators of θ ± , and (b) where (1 ± γ ) vanishes in the nu-merators. Case (a) corresponds to singular contributionfrom the entire vortex lattice (i.e. from the single termswith G = 0), whereas case (b) corresponds to contri-butions from the entire reciprocal vortex lattice, that islocal in the direct vortex lattice. At the singular points, γ = +1 , −
1, the exponential factor e − in F τ − → e − πinn F ,contributing only purely harmonic terms to the SC freeenergy in the dHvA frequency F = n F H .Slightly away from the singular points that are localin the direct vortex lattice, i.e. corresponding to manyUmklapp scattering channels, there are significant con-tributions to the SC free energy which deviate markedlyfrom harmonic behavior. They originate from G-vectorssatisfying: | G | ≈ √ n F , namely having length closeto the Fermi surface diameter. Furthermore, due to theincommensurability of the large circular Fermi surfacewith the fine polygonal vortex lattice, this Fermi surfaceresonance condition yields erratic jumps of I ν as a func-tion of n F . The final result for the first harmonic (i.e. τ − = 2 π ) of the thermodynamic potential, Ω (1 h ) SC , modu-lated by umklapp scattering effects in the vortex lattice,up to fourth order in e ∆ , takes the form: Ω (1 h ) SC / Ω (1 h ) N ’ − (cid:0) π / / √ n F (cid:1) e ∆ + [1 + w ( n F )] (cid:0) π / n F (cid:1) e ∆ , whereΩ (1 h ) N is the corresponding normal state quantity, and w ( n F ), shown in Fig.(2), represents highly anharmoniceffects of the umklapp scattering by the vortex lattice.The influence of vortex-lattice disorder is of importancenear H c where random defects which pin flux lines,and/or SC fluctuations, introduce disorder to the vortexlattice. The structure factor, averaged over the disorderrealizations in the white-noise limit, reduces to its singu-lar value, and the SC free energy up to fourth order in e ∆ , is purely harmonic, so that Ω (1 h ) SC is obtained with w ( n F ) →
0, i.e. very close to the well known Maki-Stephen expression [15, 16], as expanded to the sameorder in ∆ .The great advantage of the perturbation approach justdescribed is in the ability to derive analytical expressions,at least for the leading terms, with sensitivity to the An-dreev scattering in the vortex core regions. On the otherhand, at any order of the perturbation expansion theexpected broadening effect of the LL is absent. To seewhether the predicted erratic oscillation effect survivesthis broadening we have calculated the quasi-particleDOS at various integer values of n F by numerically solv-ing Eqs.(1). The quantum oscillation (QO) amplitudeobtained from the resulting DOS, D n F ( ε ), by means ofthe expression: D ( n F , T ) ≈ (cid:16) m ∗ ~ (cid:17) R dε D nF ( ε ) X cosh ( εX ) , derivedin the low temperature limit, X ≡ ~ ω c k B T (cid:29)
1, of the well-known formula for the thermodynamic DOS [11], is com-pared in Fig.(2) with the oscillatory modulation function w ( n F ). The good agreement between major features inthe modulated envelopes of oscillations obtained in thesecalculations confirms the conjectured relation betweenthe enhancement of the DOS slop at zero energy and D n F
40 60 80 100 w n F
40 60 80 100
FIG. 2: Top panel: Calculated thermodynamic DOS [11], as afunction of integer n F , obtained at ~ ω c / k B T = 10, from theBdG equations. Bottom panel: QO amplitude calculated atthe same n F values using perturbation GGL theory. Note thepronounced jump from n F = 57 to 58 which correlates witha sharp increase of the Dirac-shape slop shown in Fig.(1). the Fermi surface resonance-scattering through the vor-tex lattice cores. The most pronounced feature appearingquite similarly in both calculations is the sharp rise of theQO amplitude seen in Fig.2 in going from n F = 57 to 58,which is correlated with a sharp increase in the slop ofthe DOS at ε = 0 (see Fig.1). The corresponding dis-persion relation for n F = 57 (see SM1 [10]) preserves theideal conical intersection at the vortex core which char-acterizes the diagonal LL branches ± | ∆ n,n ( k ) | . Note,however, that the Dirac-shaped DOS appearing at inte-ger n F values is determined by the entire landscape ofavoiding crossings close to conical intersections appear-ing in the 2D BZ rather than by a single Dirac cone atthe vortex core (see SM1 [10]).Experimental observation of the predicted effect is ex-pected in strongly type-II, layered superconductors inwhich magneto- quantum oscillations can be observedin the mixed state [3, 17]. The only relevant exampleknown so far is κ − ( ET ) Cu ( SCN ) , which was stud-ied rather intensively by several groups[18],[19]. Suffi-ciently small interlayer transfer integral (i.e. about 0 . .
13 meV at H = 4 T, see below), ensuring 2Delectron dynamics, and clear dHvA oscillations in the SCmixed state with significant SC-induced extra damping,have been reported for this material. A modified ver-sion of the Maki-Stephen relaxation time approximation[15, 16], which takes into account the effect of SC fluctua-tions in the vortex liquid state[3] has shown a very goodquantitative agreement with the observed data [3],[21].The best fitting value of the mean-field H c ( T →
0) ob-tained in this analysis was 4 . H ∼ . − . T ). Ignoring possible quantum fluctu-ation effects this can occur at sufficiently low temper-atures where thermal fluctuations are suppressed. Thedata reported in Ref.([18]) (measured at T = 20 mK)seems to indicate that an ordered 2D vortex lattice in-deed appears in the field range H ≈ . − . T . Tosubstantiate this statement we have calculated the quasi-particle DOS using the BdG equations for a 2D vortexlattice with the field-dependent order-parameter ampli-tude calculated in the mean-field approximation (which isa good approximation in the field range investigated [3])and with parameters adopted from Ref.[18] and from thebest fit reported in Ref.([3]) (see SM2-[10]). The thermo-dynamic DOS oscillation was calculated by first convolut-ing the quasi-particle DOSs with a Lorenzian of width γ determined by the measured Dingle temperature in thenormal state to take into account the effect of atomic-lattice disorder. In the relevant field range the order-parameter amplitude is large (i.e. e ∆ ∼
3) so that theLandau bands fill essentially the entire cyclotron ‘gaps’(the broad-bands region). Two pairs of DOSs calculatedat n F = 155 , . n F = 161 , .
5, bordering thefield range of interest, are shown in Fig.(3). In the effec-tive energy interval (2 γ ) around the Fermi surface whichdominates the QO, the DOS reduction in going from n F = 155 to 155 . .
5, though in both cases the bands essentiallyfill the entire cyclotron energy interval. Thus, in the re-gion of broad Landau bands, i.e. for n F values aboveabout n F = 155, the field modulation of the thermody-namic DOS is controlled by the quasi-particle scatteringtrajectories crossing vortex-lattice cores, rather than bythe usual crossing of Landau tubes through the Fermisurface. The resulting calculated QO pattern, shown asinset in Fig.(4) as a function of 1 /H , exhibits a crossoverfrom regular oscillations, with a monotonically decreas-ing amplitude, to a ’irregularly’ modulated pattern, re-flecting the sharp field-modulation of the quasi-particleDOS under vortex-lattice cores scattering. The calcula-tion reproduces reasonably well a crossover of the sametype, starting at about H = 4 .
35 T ( n F = 156) in theexperimental data presented in Fig.(4).Some readers might argue that the sharp features seenin the experiment are just noise. It should be stressed,however, that the magnitudes of the calculated features(see, e.g. the sharp changes of the DOS with n F , andtheir fingerprints on the magneto-oscillations shown inSM2 [10] between n F = 155 and 157) are seen to be com-parable to those seen in the experimental data (Fig.(4)).It is therefore plausible that the experimentally observedfeatures arise from the predicted effect.It should be also noted that a similar calculation done n F (cid:61) Ε(cid:142) n F (cid:61) . (cid:45) (cid:45) Ε(cid:142) n F (cid:61) n F (cid:61) . (cid:45) (cid:45) FIG. 3: DOSs, as functions of e ε ≡ ε/ ~ ω c , calculated by solvingthe BdG equations 1 in the broad-band region ( e ∆ ∼
3) at n F = 155 (left panel, solid line) and 155 . n F = 161 (right panel, solid line) and 161 . H c ≈ . T ( n F = 145) at T = 20 mK reported in Ref.[18]. Thedashed line extrapolates the normal state oscillation to theSC region. The inset represents calculated QO obtained bysolving the BdG equations for a square vortex lattice as de-scribed in the text. Note the same scale on orizontal axes( n F ) for the main figure and the inset. for the hexagonal vortex-lattice model (not shown) hasyielded in this broad-bands region smoother features ascompared to the square-lattice calculation. In the nar-row Landau bands region sharp features essentially sim-ilar (but different in their details) to those shown inFigs.1,2 were obtained for the hexagonal lattice model.In conclusion, we have revealed a fundamental rela-tionship between Fermi surface resonance quasi-particlescatterings through vortex lattice cores and formation ofpronounced Dirac Fermion structures in the reciprocalvortex lattice of a 2D strongly type-II superconductor athigh magnetic fields. The predicted effect can be detectedby finely tuning the external magnetic field through res-onant Andreev scattering channels which sharply mod-ulate the quasi-particle density of states. The effect isshown to leave an observable fingerprint on magneto-quantum oscillations in the vortex-lattice state of a quasi2D organic superconductor and could be directly de-tected in future scanning tunnelling spectroscopy mea-surements. Since vortex-lattice disorder is expected tosuppress the effect [14] its observation could be usedto identify the freezing transition into the vortex-latticephase.This research was supported by E. and J. Bishop re-search fund at Technion, and by EuroMagNET under theEU contract No. 228043. V.Z. acknowledges the supportof the Israel Science Foundation by Grant No. 249/10 ∗ Electronic address: e-mail:[email protected][1] S. Dukan and Z. Tesanovic,”Superconductivity in a highmagnetic field: Excitation spectrum and tunneling prop-erties”, Phys. Rev. B , 13017 (1994).[2] S. Dukan and Z. Tesanovic,”de Haas–van Alphen Os-cillations in a Superconducting State at High MagneticFields”, Phys. Rev. Lett. , 2311 (1995).[3] T. Maniv, V. Zhuravlev, I. D. Vagner, and P.Wyder,”Vortex states and quantum magnetic oscillationsin conventional type-II superconductors”, Rev. Mod.Phys. , 867 (2001).[4] J. Wosnitza, ”Fermi surfaces of Low-Dimensional Or-ganic Metals and Superconductors”, Springer: Berlin(1996).[5] Z.Tesanovic and P. Sacramento, ”Landau Levels andQuasiparticle Spectrum of ExtremeType-II Superconductors”, Phys. Rev. Lett. , 1521(1998).[6] A. Kitaev, ”Fault-tolerant quantum computation byanyons”, Ann. Phys. , 2 (2003).[7] C. Nayak, S.H. Simon, A. Stern, M. Freedman and S.D.Sarma, ”Non-Abelian anyons andtopological quantum computation”, Rev. Mod. Phys. ,1083 (2008).[8] Suchitra E. Sebastian, Neil Harrison and G.G. Lonzarich,”Towards resolution of the Fermi surface in underdopedhigh-Tc superconductors”, Rep. Prog. Phys. , 102501(2012). [9] M. R. Norman, A.H. MacDonald, and H. Akera, Phys.Rev. B , 5927 (1995).[10] See Supplemental Material athttp://link.aps.org/supplemental/..../PhysRevLett......for details of the calculation relevant to the experimentaldata on κ -(BEDT-TTF) Cu(NCS) .[11] M. I. Katsnelson, ”Graphen, Carbon in two Dimensions”,Cambridge University Press, Cambridge (2012).[12] B. Rosenstein, M. Lewkowicz and T. Maniv, ”ChiralAnomaly and Strength of the Electron-Electron Interac-tion in Graphene”, Phys. Rev. Lett. , 066602 (2013).[13] M. R. Norman, A.H. MacDonald,”Absence of persistentmagnetic oscillations in type-II superconductors”, Phys.Rev. B , 4239 (1996).[14] V. Zhuravlev and T. Maniv, ”Exact analytic Gorkov-Ginzburg-Landau theory of type-II superconductivity inthe magnetoquantum oscillations limit”, Phys. Rev. B , 104528 (2012).[15] K. Maki, ”Quantum oscillation in vortex states of type-IIsuperconductors”, Phys. Rev. B , 2861 (1991).[16] M. J. Stephen,”Superconductors in strong magneticfields: de Haas–van Alphen effect”, Phys. Rev. B ,5481 (1992).[17] T. J. B. M. Janssen, C. Haworth, S. M. Hayden, P. Mee-son, M. Springford, and A. Wasserman,”Quantitative in-vestigation of the de Haas-van Alphen effect in the su-perconducting state”, Phys. Rev. B , 11698 (1998).[18] P.J. van der Wel, J. Caulfield, S.M. Hayden, J. Singleton,M. Springford, P. Meeson, W. Hayes, M. Kurmoo, P. Day,Synth. Met. , 831 (1995).[19] T. Sasaki, W. Biberacher, K. Neumaier, W. Hehn,K. Andres, T. Fukase,”Quantum liquid of vortices inthe quasi-two-dimensional organic superconductor σ -(BEDT-TTF)) Cu(NCS)) ”, Phys. Rev. B , 037001-1 (2002).[21] T. Sasaki, T. Fukuda, N. Yoneyama, and N. Kobayashi,”Shubnikov–de Haas effect in the quantum vortex liq-uid state of the organic superconductor κ -(BEDT-TTF) Cu(NCS) ”, Phys. Rev. B , 144521 (2003). ield-controlled conical intersections in the vortex lattice of quasi 2D pure stronglytype-II superconductors at high magnetic fields: Supplemental material T. Maniv and V. Zhuravlev
Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel (Dated: November 19, 2018)
Supplemental material 1: Landau band-structure in the Brillouin zone and Dirac-shaped density of states athalf integer filling factors
In this supplemental material we illustrate how the complex Landau band structures obtained from the BdGequations (Eq.(1) in the main text) for relatively large integer values of n F transform into the simple Dirac-shapedensity of states (DOS) functions, shown e.g. in Fig.(1) of the main text. We also show here how the landscapecharacterizing such a band structure can be controlled by varying the Landau-level filling factor, F/H +1 / n F +1 / H . This is illustrated here for the step from n F = 57 to 58 where the mostpronounced jump in the quantum-oscillation amplitude is seen in Fig.(2) of the main text. Starting with a 3D plot ofthe Landau band-structure in the entire BZ for n F = 58 shown in Fig.(1) (top-right panel) it is seen to have a veryrich landscape decorated with many avoiding crossings close to conical intersections at zero energy (mostly with non-circular directrices), which reflect the influence of the vortex-lattice cores singularity. The linear energy dependenceof the resulting DOS function near zero energy, shown in Fig.(1) of the main text, is determined by integrating overall these ’conical intersections’. Note, however, that at the vortex core position the band structure for n F = 58 hasno Dirac cone but a crater at the top of a large peak (Fig.(1) bottom-right panel). Upon varying n F from 58 to 57the high energy states contributing to this peak are shifted away from the vortex core, clearing the landscape for theDirac cone at the vortex core to be visible (see Fig.(1) left panels and Fig.(2) ). k y (cid:206)(cid:142) k (cid:206)(cid:142) FIG. 1: Top panels: Landau band-structures for n F = 57 (left) and n F = 58 (right), calculated for the square Abrikosov vortexlattice in the entire 2D BZ (note the inverted energy axes). Bottom panels: 2D plots along symmety lines passing throughthe vortex core: For n F = 57 along the k y -axis (left panel), whereas for n F = 58 along the diagonal axis (right panel). Theamplitude of the pair-potential was selected to be: ∆ ( H ) / ~ ω c = 1. a r X i v : . [ c ond - m a t . s up r- c on ] M a y FIG. 2: The energy dispersion in a restricted range around the BZ center (reciprocal vortex-lattice core position) for n F = 57,showing an extension of the central part of the top-left panel in Fig.(1) with a clear Dirac cone at the vortex core position. Supplemental material 2: Comparison with experimental dHvA oscillations data
To compare our theory with the available low temperature (20 mK) experimental data of dHvA oscillations in thevortex state of the quasi 2D organic superconductor κ − ( ET ) Cu ( SCN ) [1] we use the Abrikosov lattice form forthe pair potential [2]: ∆ ( r ) = (cid:18) πa x (cid:19) / ∆ e ixy X k e − iθk + iq k x − ( y + q k / , (1) q k = 2 πka x , k = 0 , ± , ..., a x = π/ q − ( θ/π ) where the coordinates r are measured in units of the magnetic length a H = p c ~ /eH and a x is the lattice constantalong the main principal axis. In Eq.(1) θ = 0 corresponds to the square lattice whereas θ = π/ we use the simple mean-field (BCS) form:∆ ( n F ) ~ ω c = 1 . (cid:18) T c [ K ] H [ T ] (cid:19) (cid:18) m ∗ c m (cid:19) n F (cid:18) − F/H c n F (cid:19) (2)where m ∗ c is the cyclotron effective mass, m the free electron mass and F = n F H is the dHvA frequency. For thedetected signal in the vortex state [1] F = 680 T, (cid:16) m ∗ c m (cid:17) = 3 . H c which best fits the extra dampingin the vortex liquid state is 4 . D n F ( ε ), obtained from the solutions of Eq.(1) in the main text at various values of n F is convoluted with a Lorenzian of width γ = 2 k B πT D , determined by the measured Dingle temperature T D in thenormal state, to take into account the effect of atomic-lattice disorder in the simple relaxation time approximation,i.e.: D γn F ( ε ) = 1 π Z dε D n F ( ε ) e γ ( ε − ε ) + e γ , e γ ≡ γ ~ ω c The thermodynamic DOS (or quantum capacitance) oscillation is finally calculated by means of the expression: D ( n F , T ; γ ) ≈ (cid:18) m ∗ ~ (cid:19) Z dε X cosh ( εX ) D γn F ( ε ) (3)derived in the low temperature limit, X ≡ ~ ω c k B T (cid:29)
1, of the well-known formula for the thermodynamic DOS [3].The value of T D determined from the normal state dHvA oscillation damping [5] yields γ = 0 .
28 K , that is e γ ≈ . H = 4 . FIG. 3: Calculated D ( n F , T ; γ ) in the range n F = 151 −
164 ( H = 4 . − .
15 T) (Central panel), showing sharp amplitudemodulation between n F = 155 and n F = 157 associated with the suppression of the Dirac-shaped DOS shown in the upper-leftpanel (i.e. for n F = 155) upon moving to n F = 157 (upper-right panel). half-integer values of n F is shown in Fig.(3). DOS plots in the effective quasi-particle energy range of − e γ . ε . e γ for n F = 155,155 . n F = 157, 157 .
5. The sharp reduction of the DOS in goingfrom n F = 155 to 157 and to 155 . n F = 157 to 157 . [1] P.J. van der Wel, J. Caulfield, S.M. Hayden, J. Singleton, M. Springford, P. Meeson, W. Hayes, M. Kurmoo, P. Day, Synth.Met. , 831 (1995).[2] T. Maniv, V. Zhuravlev, I. D. Vagner, and P. Wyder,”Vortex states and quantum magnetic oscillations in conventionaltype-II superconductors”, Rev. Mod. Phys. , 867 (2001).[3] M. I. Katsnelson, ”Graphen, Carbon in two Dimensions”, Cambridge University Press, Cambridge (2012).[4] T. Sasaki, T. Fukuda, N. Yoneyama, and N. Kobayashi, ”Shubnikov–de Haas effect in the quantum vortex liquid state ofthe organic superconductor κ -(BEDT-TTF) Cu(NCS) ”, Phys. Rev. B , 144521 (2003).[5] T. Sasaki, W. Biberacher, K. Neumaier, W. Hehn, K. Andres, T. Fukase,”Quantum liquid of vortices in the quasi-two-dimensional organic superconductor κ -(BEDT-TTF) Cu(NCS) ”, Phys. Rev. B57